Clustered sparsity and Poisson-gap sampling

Abstract

Non-uniform sampling (NUS) is a popular way of reducing the amount of time taken by multidimensional NMR experiments. Among the various non-uniform sampling schemes that exist, the Poisson-gap (PG) schedules are particularly popular, especially when combined with compressed-sensing (CS) reconstruction of missing data points. However, the use of PG is based mainly on practical experience and has not, as yet, been explained in terms of CS theory. Moreover, an apparent contradiction exists between the reported effectiveness of PG and CS theory, which states that a “flat” pseudo-random generator is the best way to generate sampling schedules in order to reconstruct sparse spectra. In this paper we explain how, and in what situations, PG reveals its superior features in NMR spectroscopy. We support our theoretical considerations with simulations and analyses of experimental data from the Biological Magnetic Resonance Bank (BMRB). Our analyses reveal a previously unnoticed feature of many NMR spectra that explains the success of ”blue-noise” schedules, such as PG. We call this feature “clustered sparsity”. This refers to the fact that the peaks in NMR spectra are not just sparse but often form clusters in the indirect dimension, and PG is particularly suited to deal with such situations. Additionally, we discuss why denser sampling in the initial and final parts of the clustered signal may be useful.

Revisiting aliasing noise to build more robust sparsity in nonuniform sampling 2D‐NMR

“…Aliasing artifacts in NUS spectra have been extensively characterized [14][15][16][17][18] and can be treated by oversampling [15], hypercomplex NUS [18], off-grid NUS [16,19], shifting artifacts [14,20], or by incorporating uniform tracts (a.k.a. bursts) in the sampling schedule [15].…”

Deeper Insight into Photopolymerization: The Synergy of Time-Resolved Nonuniform Sampling and Diffusion NMR

“…The sampling schedule was generated using a sampling generator in the MDDNMR package with shuffle mode and the relaxation weighting parameter T2 set to 0.07. The effect of the choice of the nonuniform sampling scheme type (e.g., Poisson-gap or random) in the case of spectra of high sparsity (as in this work) should be negligible …”

“…The effect of the choice of the nonuniform sampling scheme type (e.g., Poisson-gap or random) in the case of spectra of high sparsity (as in this work) should be negligible. 40 Data Analysis. The TR-NUS spectra were reconstructed by the TReNDS program using the IRLS algorithm 41 with virtual echo.…”

NMR spectrum reconstruction as a pattern recognition problem

“…1. Implication of the different sampling schemes for appearance of the PSF and for the outcome of the spectra reconstruction using traditional techniques have been previously discussed previously [32]. Specifically, the Poisson-gap schedules tend to push the aliasing artefacts away from the main peak and thus benefit spectra with clustered signals.…”